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Binding energy is the mechanical
energy required to disassemble a whole into separate parts. A
bound system has typically a lower potential energy than its constituent
parts; this is what keeps the system together. The usual convention
is that this corresponds to a positive binding energy.

In general, binding energy represents the mechanical work which must be done in
acting against the forces which hold an object together, while
disassembling the object into component parts separated by
sufficient distance that further separation requires negligible
additional work.

At the nuclear level the nuclear
binding energy (binding energy of nucleons into a nuclide) is derived from the strong nuclear force and is the energy required to disassemble a
nucleus into
the same number of free unbound neutrons and protons it is composed of, in such a way that
the particles are far/distant enough from each other so that the
strong nuclear force can no longer cause the particles to
interact.

In astrophysics, gravitational binding
energy of a celestial body is the energy required to expand the
material to infinity. This quantity is not to be confused with the
gravitational potential energy, which is
the energy required to separate two bodies, such as a celestial
body and a satellite, to infinite distance, keeping each intact
(the latter energy is lower).

In bound systems, if the binding energy is removed from the
system, it must be subtracted from the mass of the unbound system,
simply because this energy has mass, and if subtracted from the
system at the time it is bound, will result in removal of mass from
the system. System mass is not conserved in this process because
the system is not closed during the binding
process.

Mass
deficit

Classically a bound system is at a lower energy level than its
unbound constituents, its mass must be less than the total mass of
its unbound constituents. For systems with low binding energies,
this "lost" mass after binding may be fractionally small. For
systems with high binding energies, however, the missing mass may
be an easily measurable fraction.

Since all forms of energy have mass, the question of where the
missing mass of the binding energy goes, is of interest. The answer
is that this mass is lost from a system which is not closed. It
transforms to heat, light, higher energy states of the nucleus/atom
or other forms of energy, but these types of energy also have mass,
and it is necessary that they be removed from the system before its
mass may decrease. The "mass deficit" from binding energy is
therefore removed mass that corresponds with removed energy,
according to Einstein's equation E = mc2. Once
the system cools to normal temperatures and returns to ground
states in terms of energy levels, there is less mass remaining in
the system than there was when it first combined and was at high
energy. Mass measurements are almost always made at low
temperatures with systems in ground states, and this difference
between the mass of a system and the sum of the masses of its
isolated parts is called a mass deficit. Thus, if binding energy
mass is transformed into heat, the system must be cooled (the heat
removed) before the mass-deficit appears in the cooled system. In
that case, the removed heat represents exactly the mass "deficit",
and the heat itself retains the mass which was lost.

As an illustration, consider two objects attracting each other
in space through their gravitational field. The attraction
force accelerates the objects and they gain some speed toward each
other converting the potential (gravity) energy into kinetic
(movement) energy. When either the particles 1) pass through each
other without interaction or 2) elastically repel during the
collision, the gained kinetic energy (related to speed), starts to
revert into potential form driving the collided particles apart.
The decelerating particles will return to the initial distance and
beyond into infinity or stop and repeat the collision (oscillation
takes place). This shows that the system, which loses no energy,
does not combine (bind) into a solid object, parts of which
oscillate at short distances. Therefore, in order to bind the
particles, the kinetic energy gained due to the attraction must be
dissipated (by resistive force). Complex objects in collision
ordinarily undergo inelastic collision, transforming
some kinetic energy into internal energy (heat content, which is
atomic movement), which is further radiated in the form of
photons—the light and heat. Once the energy to escape the gravity
is dissipated in the collision, the parts will oscillate at closer,
possibly atomic, distance, thus looking like one solid object. This
lost energy, necessary to overcome the potential barrier in order
to separate the objects, is the binding energy. If this binding
energy were retained in the system as heat, its mass would not
decrease. However, binding energy lost from the system (as heat
radiation) would itself have mass, and directly represent of the
"mass deficit" of the cold, bound system.

Closely analogous considerations apply in chemical and nuclear
considerations. Exothermic chemical reactions in closed systems do
not change mass, but (in theory) become less massive once the heat
of reaction is removed. This mass change is too small to measure
with standard equipment. In nuclear reactions, however, the
fraction of mass that may be removed as light or heat, i.e.,
binding energy, is often a much larger fraction of the system mass.
It may thus be measured directly as a mass difference between rest
masses of reactants and products. This is because nuclear forces
are comparatively stronger than Coulombic forces associated with
the interactions between electrons and protons that generate heat
in chemistry.

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Mass
defect

In simple words definition of mass defect can be stated as
follows:

Definition: The difference between the unbound system calculated
mass and experimentally measured mass of nucleus is called mass
defect. It is denoted by Δm. It can be calculated as follows:

i.e, (sum of masses of protons and neutrons) - (measured mass
of nucleus)

In nuclear reactions, the energy that must be radiated or otherwise
removed as binding energy may be in the form of electromagnetic
waves, such as gamma radiation, or as heat. Again,
however, no mass deficit can in theory appear until this radiation
has been emitted and is no longer part of the system.

The energy given off during either nuclear fusion or nuclear fission
is the difference between the binding energies of the fuel and the
fusion or fission products. In practice, this energy may also be
calculated from the substantial mass differences between the fuel
and products, once evolved heat and radiation have been
removed.

When the nucleons are grouped together to form a nucleus, they
lose a small amount of mass, i.e., there is mass defect. This mass
defect is released as (often radiant) energy according to the
relation E = mc2; thus binding energy
= mass defect × c2.

This energy is a measure of the forces that hold the nucleons
together, and it represents energy which must be supplied from the
environment if the nucleus is to be broken up. It is known as
binding energy, and the mass defect is a measure of the binding
energy because it simply represents the mass of the energy which
has been lost to the environment after binding.

Mass
excess

It is observed experimentally that the mass of the nucleus is
smaller than the number of nucleons each counted with a mass of 1
a.m.u..
This difference is called mass excess.

The difference between the actual mass of the nucleus
measured in atomic mass units and the number of
nucleons is called mass excess i.e.

with : M equals the actual mass of the nucleus, in u.
and : A equals the mass number.

This mass excess is a practical value calculated from
experimentally measured nucleon masses and stored in nuclear
databases. For middle-weight nuclides this value is negative in
contrast to the mass defect which is never negative for any
nuclide.

Nuclear
binding energy

Practice: Binding energy
for atoms

Definition: The amount of energy required to break the nucleus
of an atom into its isolated nucleons is called Nuclear binding
energy. The measured mass deficits of isotopes are always listed
as mass deficits of the neutralatoms of that isotope, and mostly in MeV. As a
consequence, the listed mass deficits are not a measure for the
stability or binding energy of isolated nuclei, but for the whole
atoms. This has very practical reasons, because it is very hard to
totally ionize heavy elements, i.e.
strip them of all of their electrons.

This practice is useful for other reasons, too: Stripping all
the electrons from a heavy unstable nucleus (thus producing a bare
nucleus) will change the lifetime of the nucleus, indicating that
the nucleus cannot be treated independently (Experiments at the
heavy ion accelerator GSI). This is
also evident from phenomena like electron capture. Theoretically, in
orbital models of heavy atoms, the electron orbits partially inside
the nucleus (it doesn't orbit in a strict sense, but has a
non-vanishing probability of being located inside the nucleus).

Of course, a nuclear decay happens to the nucleus,
meaning that properties ascribed to the nucleus will change in the
event. But for the following considerations and examples, you
should keep in mind that "mass deficit" as a measure for "binding
energy", and as listed in nuclear data tables, means "mass deficit
of the neutral atom" and is a measure for stability of the whole
atom.

Nuclear binding energy
curve

Binding energy per nucleon of common isotopes.

In the periodic
table of elements, the series of light elements from hydrogen up to sodium is observed to exhibit
generally increasing binding energy per nucleon as the atomic mass increases.
This increase is generated by increasing forces per nucleon in the
nucleus, as each additional nucleon is attracted by all of the
other nucleons, and thus more tightly bound to the whole.

The region of increasing binding energy is followed by a region
of relative stability (saturation) in the sequence from magnesium through xenon. In this region, the nucleus
has become large enough that nuclear forces no longer completely
extend efficiently across its width. Attractive nuclear forces in
this region, as atomic mass increases, are nearly balanced by
repellent electromagnetic forces between protons, as atomic number
increases.

Finally, in elements heavier than xenon, there is a decrease in
binding energy per nucleon as atomic number increases. In this
region of nuclear size, electromagnetic repulsive forces are
beginning to gain against the strong nuclear force.

At the peak of binding energy, nickel-62 is the most tightly-bound nucleus
(per nucleon), followed by iron-58 and iron-56.[1] This is
the approximate basic reason why iron and nickel are very common
metals in planetary cores, since they are produced profusely as end
products in supernovae
and in the final stages of silicon burning in
stars. However, it is not binding energy per defined nucleon (as
defined above) which controls which exact nuclei are made, because
within stars, neutrons are free to convert to protons to release
even more energy, per generic nucleon, if the result is a stable
nucleus with a larger fraction of protons. Thus, iron-56 has most
binding energy of any group of 56 nucleons (because of its
relatively larger fraction of protons), even while having less
binding energy per nucleon than nickel-62, if this binding energy
is computed by comparing Ni-62 with its disassembly products of 28
protons and 34 neutrons. In fact, it has been argued that photodisintegration of
62Ni to form 56Fe may be energetically
possible in an extremely hot star core, due to this beta decay
conversion of neutrons to protons.[2]

It is generally believed that iron-56 is more common than nickel
isotopes in the universe for mechanistic reasons, because its
unstable progenitor nickel-56 is copiously made by staged
build-up of 14 helium nuclei inside supernovas, where it has no
time to decay to iron before being released into the interstellar
medium in a matter of a few minutes as a star explodes. However,
nickel-56 then decays to iron-56 within a few weeks. The gamma ray light
curve of such a process has been observed to happen in type IIa
supernovae, such as SN1987a. In a star, there are no good ways
to create nickel-62 by alpha-addition processes, or else there
would presumably be more of this highly-stable nuclide in the
universe.

Measuring the binding
energy

The existence of a maximum in binding energy in
medium-sized nuclei is a consequence of the trade-off in the
effects of two opposing forces which have different range
characteristics. The attractive nuclear force (strong nuclear force), which binds protons
and neutrons equally to each other, has a limited range due to a
rapid exponential decrease in this force with distance. However,
the repelling electromagnetic force, which acts between protons to
force nuclei apart, falls off with distance much more slowly (as
the inverse square of distance). For nuclei larger than about four
nucleons in diameter, the additional repelling force of additional
protons more than offsets any binding energy which results between
further added nucleons as a result of additional strong force
interactions; such nuclei become less and less tightly bound as
their size increases, though most of them are still stable.
Finally, nuclei containing more than 209 nucleons (larger than
about 6 nucleons in diameter) are all too large to be stable, and
are subject to spontaneous decay to smaller nuclei.

Nuclear
fusion produces energy by combining the very lightest elements
into more tightly-bound elements (such as hydrogen into helium), and nuclear fission
produces energy by splitting the heaviest elements (such as uranium and plutonium) into more
tightly-bound elements (such as barium and krypton). Both processes produce energy,
because middle-sized nuclei are the most tightly bound of all.

As seen above in the example of deuterium, nuclear binding
energies are large enough that they may be easily measured as
fractional mass deficits,
according to the equivalence of mass and energy. The atomic binding
energy is simply the amount of energy (and mass) released, when a
collection of free nucleons are joined together to form a nucleus.

Nuclear binding energy can be easily computed from the easily
measurable difference in mass of a nucleus, and the sum of the
masses of the number of free neutrons and protons that make up the
nucleus. Once this mass difference, called the mass
defect or mass deficiency, is known,
Einstein's mass-energy
equivalence formula E = mc² can be
used to compute the binding energy of any nucleus. (As a historical
note, early nuclear physicists used to refer to computing this
value as a "packing fraction" calculation.)

For example, the atomic mass unit (1
u) is defined to be 1/12 of the mass of a
12C atom—but the atomic mass of a 1H atom
(which is a proton plus electron) is 1.007825 u,
so each nucleon in 12C has lost, on average, about 0.8%
of its mass in the form of binding energy.

Semiempirical
formula for nuclear binding energy

For a nucleus with A nucleons including Z protons, a
semiemipirical formula for the binding energy
(B.E.) per nucleon (A) is:

where the binding energy is in MeV for the following numerical
values of the constants: a =
14.0; b = 13.0; c = 0.585; d = 19.3; e = 33.

The first term
is called the saturation contribution and ensures that the B.E. per
nucleon is the same for all nuclei to a first approximation. The
term
is a surface tension effect and is proportional to the number of
nucleons that are situated on the nuclear surface; it is largest
for light nuclei. The term
is the Coulomb electrostatic repulsion; this becomes more important
as Z increases. The symmetry
correction term
takes into account the fact that in the absence of other effects
the most stable arrangement has equal numbers of protons and
neutrons; this is because the n-p interaction in a nucleus
is stronger than either the n-n or p-p
interaction. The pairing term
is purely empirical; it is + for even-even nuclei and - for odd-odd nuclei.

All mass excess
data are taken from [3]. Notice
also that we use 1 u = 1 a.m.u = 931.494028(±0.000023) MeV. To
calculate the "binding energy" we use the formula
P*(mp+me) + N * mn -
mnuclide where P denotes the number of protons of the
nuclides and N its number of neutrons. We take mp =
938.2723 Mev, me = 0.5110 MeV and mn =
939.5656 MeV. The letter A denotes the sum of P and N (number of
nucleons in the nuclide). If we assume the reference nucleon has
the mass of a neutron (so that all "total" binding energies
calculated are maximal) we could define the total binding energy as
the difference from the mass of the nucleus, and the mass of a
collection of A free neutrons. In other words, it would be [(P+N)*
mn] - mnuclide. The "total
binding energy per nucleon" would be this value divided by A.

In this calculation 56Fe has the lowest
nucleon-specific mass of the four nuclides, but this does not mean
it is the strongest bound atom per hadron, unless the choice of
beginning hadrons is completely free. Iron releases the largest
energy if any 56 nucleons are allowed to build a nuclide—changing
one to another if necessary, The highest "binding energy" per
hadron, with the hadrons starting as the same number of protons Z
and total nucleons A as in the bound nucleus, is 62Ni.
Thus, the true absolute value of the total binding energy of a
nucleus depends on what we are "allowed" to construct the nucleus
out of. If all nuclei of mass number A were to be allowed to be
constructed of A neutrons, then Fe-56 would release the most energy
per nucleon, since it has a larger fraction of protons than Ni-62.
However, if nucleons are required to be constructed of only the
same number of protons and neutrons that they contain, then
nickel-62 is the most tightly bound nucleus, per nucleon.

In the table above it can be seen that the decay of a neutron,
as well as the transformation of tritium into helium-3, releases
energy; hence, it manifests a stronger bound new state when
measured against the mass of an equal number of neutrons (and also
a lighter state per number of total hadrons). Such reactions are
not driven by changes in binding energies as calculated from
perviously fixed N and Z numbers of neutrons and protons, but
rather in decreases in the total mass of the nuclide/per nucleon,
with the reaction.